Stephen Colbert was running through a tongue-in-cheek opening segment on Barack Obama’s executive powers when the president of the United States walked, unannounced, onto the stage at a sold-out Lisner Auditorium.

“I'm thrilled that you're here but I did not expect you for another three minutes,” the host of “The Colbert Report” deadpanned.

President Obama stepped to the middle of the stage and shook Mr. Colbert’s hand.

“You’ve been taking a lot of shots at my job,” he said. “I decided I’m going to go ahead and take a shot at yours.”

“As president you still have a lot of responsibilities,” President Obama said when Mr. Colbert playfully pointed out that it was a little surprising to see him take executive action after the Democrats were pummeled in November. “Part of my job these next two years—and hopefully the job of [congressional Republican leaders] Mitch McConnell and John Boehner—is to show that even in divided government, we can still put people ahead of politics.”

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One of those areas, he said, concerns the effect of a minimum wage increase on the domestic economy. The November jobs report, released last week, reflected 321,000 new jobs. But while job growth and the stock market are rising, President Obama said, wages are still stagnant.

“Our argument has been that when you pass minimum wage increases—when you make sure you have equal pay—those things are not only good for individual workers, they are good for the economy because when ordinary people have some money in their pocket, they spend it. And when they spend it, that means businesses have more customers and hire more people,” he said.

When the discussion turned to the Keystone XL pipeline, the president first let the audience and Mr. Colbert do the talking for him.

“Congress is going to pass that—House and Senate—the American people want it, it’s going to create jobs and the State Department says it’s not going to raise pollution in the atmosphere,” Mr. Colbert said to rising laughter. “You’re going to sign it, right?”

Pockets of boos emerged from the crowd. President Obama looked at Mr. Colbert, and then to the audience.

“No,” Mr. Colbert said. “They’re chanting ‘doooo it.’”

President Obama side-stepped the question.

"Essentially this is Canadian oil, passing through the United States to be sold on the world market,” he said. “It’s not going to push down gas prices here in the United States. It’s good for Canada, it could create a couple thousand jobs here in the U.S. in the initial construction of the pipeline, but we have to measure that against whether or not it’s going to contribute to an overall warming of the planet that could be disastrous."

On the topic of immigration, the president stood by his executive order and rebuffed Mr. Colbert’s accusation that his executive actions have established more reach than his predecessors in the Oval Office. A FiveThirtyEight data lab report listing presidential executive orders has President Obama issuing them at a lower rate per year than every president since Grover Cleveland. Detractors, meanwhile, have said his orders go further than the nuanced executive actions of past presidents.

“Weren't those previous decisions in the wake of laws that had been passed, but they changed the implementation of those laws?” Mr. Colbert said. “This is just you saying ‘I declare.’”

President Obama smiled.

“What happened in those situations was Congress passed laws and then deliberately left out some things the president wanted but they still felt it was the right thing to do …” he said before pivoting back to the issue of immigration. “What we know is the best thing America has going for us is immigration. About half of Silicon Valley’s companies were founded by immigrants.”

Still, the power of the executive has long been a recurring theme of the Obama presidency. Mr. Colbert tweaked the president on that notion a few times Monday—"You realize you're an emperor now; it has been declared"—and returned to the topic of executive power toward the end of the interview.

“There’s always the temptation to want to go ahead and get stuff done,” President Obama said. “But my preference would be to get a whole lot more done through Congress. On the immigration legislation what I said was, ‘If you don’t agree with how we’re approaching this executive action, there’s an easy solution: Pass a bill.’ Too often we have a Congress that is stuck, and the executive and the courts end up filling the gaps.”

Nearly six years after his 2009 inauguration, he said he still enjoys the roles and responsibilities of the office. His wife and two daughters, he said, keep him grounded.

“I love the job,” he said. “You’re not thinking about it in terms of titles, you’re thinking of how you can deliver for the American people. And also, when I get home, Michelle, Malia and Sasha give me a hard time. They tease me.”